Math 317 Theory of Linear Algebra Fall 2005

         New Information

Grades are posted in WebCT, along with grading information.

Class Friday Dec. 9 will be an optional computation review, in problems session format.
computation review, in problems session format, will continue Monday Dec. 12 ]time]
Office Hours
Mon Dec 12 1-4 PM (with computation review 1-2 in Carver 124)
Thursday Dec 15 9-11 AM

Final Exam info

Final exam Tuesday Dec. 13 12-2 PM in the regular classroom (4 Carver)

Index

Homework assignments
Test Information
General course information
Grades are posted in WebCT

Homework Assignments

Always read the section of the text on which problems are assigned
The problems will collected on the Monday following the week in which they are assigned (Aug. 22-Aug. 26) were collected Aug. 29 .   When handed in, homework assignments should be stapled into one packet, in order.
date section problems
M 22-Aug 1.1 1ab, 3ab, 5ab, 6ab, 7acf, 23(5)(7)
T 23 1.2 (through p. 22) 1ab, 5, 6(1), 7, 10, 11c
R 25 1.3 5, 7, 10a, and handout 1 problems HH1 - HH5
F 26 1.4  1abefij, 10
M 29 1.5 1abjk, 2ad, and 1.4: 6b, 13a
T 30 1.5 3ac, 7a, 18, 20(AA^T)
R 1-Sept 2.1 2, 3, 4
F 2 2.1 (see also 2.2) 2.1: 1ab by Gauss-Jordan (RREF), 1cf by Gaussian (REF and back-sub), 10. 11
T 6 2.2  1, 2abcd, 6a, 7a
R 8 2.3 4, 5bc, 7, 18ab; 2.2: 9, 13a
F 9 2.3 problem HH1 below
M 12
 2.4
 1a, 3ab, 4abd, 15a(1)
T 13
 2.4
 2ab, 7a, 9, 13, 18, prove Lemma 10a on extra theorem list
R 15
 3.1
 1abef, 3b, 5ab, 7
F 16
 3.2
 1, 2bcd
M 19
 3.2
 8, 11a, 12a, Prove Cor. 3.4, Prove Lemma H14 (below)
M 26
 3.3
 2ad, 5a, 8, 13bcd
T 27
 3.4
 1abc, 2abc
R 29
 3.4
 4abde, 6, 12, 17, also problem HH2 below
M 3-Oct
 4.1
 5,6,7,8,12a,14
T 4
 4.2
 1, 2, 3
R 6
 4.2
 8, 11, 12ab, 18
F 7
 4.3, 4.4
 4.3: 6, 7, 8, 9, 24; 4.4: 1, 2abcd
M 10
 4.4
 3, 4abd, 5. 6, 13, 20, 27
T11
 4.5
 1a, 2, 3, 4ace, 6
R13
 4.5
 18, 19, 22
F14
 4.6
 5ab, 7ab
M17
 4.7
 1abc, 12, prove Lemma H27
T18
 4.7
 2ab, 6
M24
 5.1
 1, 2
T25
 5.1
 8, 16, 18a, 22, 31, 33
R27
 5.2
 2, 9a, Prove Theorem H30(1)
F28
 5.2
 3, 5
T1Nov
 5.3
 1, 3, Prove Thm 5.9 without using Thm 5.3, HH3 below
R3
 5.3
 6, 15, 20
F4
 5.5
 2bc
M7
 5.5
  6, 10, 13
T8
 5.4
 1, 2, 13, 17, 18b
R10
 6.1
 1, 2; also 1.2: 15, 16
F11
 6.1
 3, 4
M14
 6.2
 1ab, see * below
T15
 6.2
 1a, 2acd, 6, 10a, 11, 12; 6.1: 5a; prove Thm H38 in Notes (material on projections)
R17
 6.3
 1abc, 2ab, 3ac, and Thm H40 below
T29
 7.5
 1, 7, HH4 below

HH4 On C[0,1], <f,g>=integral from 0 to 1 of f(x)g(x).
a) find an orthonormal basis for P1.
b) compute the projection of x^2 onto P1.


Thm H40 Let Q1, Q2 be orthogonal nxn matrices.  Then
a) Q1 Q2 is an orthogonal matrix.
b) Q1^-1 is na orthogonal matrix.

* W={(2a-2b,a+2b,2a+b) : a,b real}.  Find proj_W(1,2,3)
we will discuss this tomorrow


HH1:
Here is the homework problem assigned Sept 9:
             [ 1 2 3 ]
Let A = [ 4 5 6 ]
             [ 7 8 9 ]
Find elementary row matrices E_1, ..., E_k such that
E_k...E_1A=RREF(A)
Hint: find elementary row operations R_1,...,R_k such that doing row operations R_1 to R_k to A gives RREF(A) and let E_i=R_i(I)

HH2: Prove:
If A is an invertible matrix and r is an eigenvalue of A, then r is nonzero and r^-1 ia an eigenvalue of A^-1.

Lemma H14: If A, B are square same size and AB is invertible, then both A and B are invertible.

Lemma H27 Let B be an ordered basis for finite dimensional vectors space V and let v,w be in V.  Then
[v]_B = [w]_B if and only if v = w.

Theorem H30(1) Let A be an m x n matrix.  Then [T_A]=A.

HH3: Let M22 = 2 x 2 real matrices.  Let L: M22 -> M22 be defined by
L(B)= AB where A=[3 2].
                                  [2 6]
a) Find [L], the matrix of L with respect to the standard basis for M22
b) Find the eigenvalues and bases for the eigenspaces for [L]
c) Find the eigenvalues and bases for the eigenspaces for L
d) Find an ordered basis C of M22 such that C_[L]_C is didagonal (and find C_[L]_C).

Test Information

Final exam Tuesday Dec. 13 12-2 PM in the regular classroom (4 Carver)

Final Exam Directions:

Math 317                  Final Examination             December 13, 2005

Directions:
Answer eight questions, including at least three from each part, well (you may answer more questions but only the best 8 scores following the 4-4 or 5-3 or 3-5 distribution will be used). Show your work. All proofs must be in Statement/Reason format except the following:
a)    showing a function is/is not a linear transformation
b)    showing a subset is/is not a subspace
c)    an induction proof. 
You may use your pre-approved theorem list and may cite any correct result on it unless explicitly disallowed by the question.  All answers must be justified by computation or explanation.  You may use a calculator, and no work is required for reduction to reduced row echelon form (RREF) or for real arithmetic.  For all other computations you must show work that justifies the answer without a calculator or be able to do it in your head.    denotes the real numbers.   denotes  real matrices, etc.

Part I: Computations

Part II: Proofs


Test 3 solution posted outside 488 Carver




Old Test 3
Test 2               test 2 solutions
old Test 2         old test 2 solutions

old Test 1         old test 1 solutions

Updated (22-Nov-05) H-theorem list

The first test was Thursday September 22.  The test covered Chapters 1 and 2. 

Test 2 will be Thursday, Oct. 20 covering Chapters 3 and 4.

Test 3 will by Friday Dec. 2 covering Chapters 5 and 6.

You may use a theorem list that you prepare yourself on 1 additional standard sheet of paper (front and back ok, can be typed or hand-written).  You can reuse your test 1 and test 2 lists for test 3.  Theorem lists cannot have examples or definitions and must be your own work.  It can include any theorem, lemma, proposition, or corollary in the text or on the extra theorem list (H list).  Submit your lists in class Tuesday Nov. 30 to get pre-approval.

Class Time and Place

        MTRF 12:10-1 PM in 4 Carver

Text

Course Content

This course emphasizes the reading and writing of mathematical proofs in addition to the content of linear algebra.  Linear Algebra topics include vectors & matrices, systems of linear equations, determinants, eigenvalues/eigenvectors, vector spaces, linear transformations, orthogonality and inner product spaces.  Most of Ch 1-6 of the text will be covered.

Polices/Syllabus Best 8 of 10 homework grades will be used for homework total score (this modifies the information in the Policy statement).  Homework total score will be normalized to 100 points possible.

Instructor Information

Disability Information Iowa State University complies with the American with Disabilities Act and Section 504 of the Rehabilitation Act.  If a student has a disability that qualifies and requires accommodations, he/she should contact the Disability Resources (DR) office for information on appropriate policies and procedures. DR is located on the main floor of the Student Services Building, Room 1076; their phone is 515-294-6624. Any student who requires an accommodation under such provisions should contact me privately as soon as possible and no later than the end of the first week of class or as soon as documentation of the need for accommodation is obtained. Contact may be made by e-mail (LHogben@iastate.edu), telephone (4-8168), or in person (office 488 Carver).  No retroactive accommodations will be provided in this class.
 
 
 
 
 
Leslie Hogben's Homepage updated after each class period